Algebra Examples

$\frac{x}{x - 1} - \frac{1}{x + 1} = \frac{2}{x^{2} - 1}$

Step 1

Subtract $\frac{2}{x^{2} - 1}$ from both sides of the equation.

$\frac{x}{x - 1} - \frac{1}{x + 1} - \frac{2}{x^{2} - 1} = 0$

Step 2

Simplify $\frac{x}{x - 1} - \frac{1}{x + 1} - \frac{2}{x^{2} - 1}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{x}{x - 1} - \frac{1}{x + 1} - \frac{2}{x^{2} - 1^{2}} = 0$

Step 2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{x}{x - 1} - \frac{1}{x + 1} - \frac{2}{(x + 1) (x - 1)} = 0$

Step 2.2

To write $\frac{x}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{x}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{1}{x + 1} - \frac{2}{(x + 1) (x - 1)} = 0$

Step 2.3

To write $- \frac{1}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{x}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{1}{x + 1} \cdot \frac{x - 1}{x - 1} - \frac{2}{(x + 1) (x - 1)} = 0$

Step 2.4

Write each expression with a common denominator of $(x - 1) (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{x}{x - 1}$ by $\frac{x + 1}{x + 1}$ .

$\frac{x (x + 1)}{(x - 1) (x + 1)} - \frac{1}{x + 1} \cdot \frac{x - 1}{x - 1} - \frac{2}{(x + 1) (x - 1)} = 0$

Step 2.4.2

Multiply $\frac{1}{x + 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{x (x + 1)}{(x - 1) (x + 1)} - \frac{x - 1}{(x + 1) (x - 1)} - \frac{2}{(x + 1) (x - 1)} = 0$

Step 2.4.3

Reorder the factors of $(x - 1) (x + 1)$ .

$\frac{x (x + 1)}{(x + 1) (x - 1)} - \frac{x - 1}{(x + 1) (x - 1)} - \frac{2}{(x + 1) (x - 1)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{x (x + 1) - (x - 1)}{(x + 1) (x - 1)} - \frac{2}{(x + 1) (x - 1)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{x (x + 1) - (x - 1) - 2}{(x + 1) (x - 1)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 - (x - 1) - 2}{(x + 1) (x - 1)} = 0$

Step 2.7.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 1 - (x - 1) - 2}{(x + 1) (x - 1)} = 0$

Step 2.7.3

Multiply $x$ by $1$ .

$\frac{x^{2} + x - (x - 1) - 2}{(x + 1) (x - 1)} = 0$

Step 2.7.4

Apply the distributive property.

$\frac{x^{2} + x - x - - 1 - 2}{(x + 1) (x - 1)} = 0$

Step 2.7.5

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} + x - x + 1 - 2}{(x + 1) (x - 1)} = 0$

Step 2.8

Combine the opposite terms in $x^{2} + x - x + 1 - 2$ .

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Step 2.8.1

Subtract $x$ from $x$ .

$\frac{x^{2} + 0 + 1 - 2}{(x + 1) (x - 1)} = 0$

Step 2.8.2

Add $x^{2}$ and $0$ .

$\frac{x^{2} + 1 - 2}{(x + 1) (x - 1)} = 0$

Step 2.9

Subtract $2$ from $1$ .

$\frac{x^{2} - 1}{(x + 1) (x - 1)} = 0$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Rewrite $1$ as $1^{2}$ .

$\frac{x^{2} - 1^{2}}{(x + 1) (x - 1)} = 0$

Step 2.10.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{(x + 1) (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.11

Cancel the common factor of $x + 1$ .

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Step 2.11.1

Cancel the common factor.

$\frac{(x + 1) (x - 1)}{(x + 1) (x - 1)} = 0$

Step 2.11.2

Rewrite the expression.

$\frac{x - 1}{x - 1} = 0$

Step 2.12

Cancel the common factor of $x - 1$ .

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Step 2.12.1

Cancel the common factor.

$\frac{x - 1}{x - 1} = 0$

Step 2.12.2

Rewrite the expression.

$1 = 0$

Step 3

Since $1 \neq 0$ , there are no solutions.

No solution